Question

By selling an article for Rs. 48, a trader loses as much percent as half of the cost price of the article. Calculate the cost price and loss amount of the article.

Hint:

Always test both positive roots of a quadratic equation in word problems.
Often, both roots are physically realistic and represent two perfectly valid solutions!

Answer 1

Step 1: Understanding the Question:
We are given the selling price (SP) of an article as Rs. 48.
The loss percent suffered by the trader is numerically equal to half of the cost price (CP) of the article.
We need to determine the cost price and the actual loss amount.

Step 2: Key Formula or Approach:
Let the cost price of the article be $x$ rupees.
The loss percentage is given as half of the cost price, i.e., $\text{Loss \%} = \frac{x}{2}\%$.
The relationship between SP, CP, and Loss % is:
\[ \text{Loss \%} = \frac{\text{CP} - \text{SP}}{\text{CP}} \times 100 \]
We will substitute our variables and solve the resulting quadratic equation.

Step 3: Detailed Explanation:

• Set up the equation:
\[ \frac{x}{2} = \frac{x - 48}{x} \times 100 \]

• Simplify the equation by cross-multiplying:
\[ x \times x = 2 \times 100 \times (x - 48) \]
\[ x^2 = 200(x - 48) \]
\[ x^2 = 200x - 9600 \]
Rearrange all terms to one side to form a quadratic equation:
\[ x^2 - 200x + 9600 = 0 \]

• Solve the quadratic equation by splitting the middle term:
We need two numbers that multiply to $9600$ and add to $-200$. These numbers are $-120$ and $-80$.
\[ x^2 - 120x - 80x + 9600 = 0 \]
\[ x(x - 120) - 80(x - 120) = 0 \]
\[ (x - 120)(x - 80) = 0 \]
This yields two possible values for the cost price:
\[ x = 120 \quad \text{or} \quad x = 80 \]

• Calculate the loss amount for both cases:
- Case 1: Cost Price (CP) = Rs. 120
Loss percentage $= \frac{120}{2}\% = 60\%$
Loss amount $= 60\% \text{ of } 120 = \frac{60}{100} \times 120 = \text{Rs. } 72$
Selling Price $= 120 - 72 = \text{Rs. } 48$ (Verified)
- Case 2: Cost Price (CP) = Rs. 80
Loss percentage $= \frac{80}{2}\% = 40\%$
Loss amount $= 40\% \text{ of } 80 = \frac{40}{100} \times 80 = \text{Rs. } 32$
Selling Price $= 80 - 32 = \text{Rs. } 48$ (Verified)


Step 4: Final Answer:
The possible solutions are:
1. Cost Price is Rs. 120 and the loss amount is Rs. 72.
2. Cost Price is Rs. 80 and the loss amount is Rs. 32.